Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
x∈(−∞,−1]∪(3,+∞)
Evaluate
x−3x+1≥0
Find the domain
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Evaluate
x−3=0
Move the constant to the right side
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
x−3x+1≥0,x=3
Set the numerator and denominator of x−3x+1 equal to 0 to find the values of x where sign changes may occur
x+1=0x−3=0
Calculate
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Evaluate
x+1=0
Move the constant to the right-hand side and change its sign
x=0−1
Removing 0 doesn't change the value,so remove it from the expression
x=−1
x=−1x−3=0
Calculate
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Evaluate
x−3=0
Move the constant to the right-hand side and change its sign
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
x=−1x=3
Determine the test intervals using the critical values
x<−1−1<x<3x>3
Choose a value form each interval
x1=−2x2=1x3=4
To determine if x<−1 is the solution to the inequality,test if the chosen value x=−2 satisfies the initial inequality
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Evaluate
−2−3−2+1≥0
Simplify
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Evaluate
−2−3−2+1
Add the numbers
−2−3−1
Subtract the numbers
−5−1
Cancel out the common factor −1
51
51≥0
Calculate
0.2≥0
Check the inequality
true
x<−1 is the solutionx2=1x3=4
To determine if −1<x<3 is the solution to the inequality,test if the chosen value x=1 satisfies the initial inequality
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Evaluate
1−31+1≥0
Simplify
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Evaluate
1−31+1
Add the numbers
1−32
Subtract the numbers
−22
Reduce the numbers
1−1
Calculate
−1
−1≥0
Check the inequality
false
x<−1 is the solution−1<x<3 is not a solutionx3=4
To determine if x>3 is the solution to the inequality,test if the chosen value x=4 satisfies the initial inequality
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Evaluate
4−34+1≥0
Simplify
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Evaluate
4−34+1
Add the numbers
4−35
Subtract the numbers
15
Divide the terms
5
5≥0
Check the inequality
true
x<−1 is the solution−1<x<3 is not a solutionx>3 is the solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
x≤−1 is the solutionx>3 is the solution
The final solution of the original inequality is x∈(−∞,−1]∪(3,+∞)
x∈(−∞,−1]∪(3,+∞)
Check if the solution is in the defined range
x∈(−∞,−1]∪(3,+∞),x=3
Solution
x∈(−∞,−1]∪(3,+∞)
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